FR3140438A1 - Method for bi-level optimization of the location of defect detection sensors in a structure using guided elastic wave tomography - Google Patents

Abstract

Method for bi-level optimization of the location of defect detection sensors in a structure by guided elastic wave tomography. The invention relates to a method for bi-level optimization of the location of defect detection sensors (s) in a tomography structure comprising an initialization/generation phase of a database comprising N reference images associated with N defects, and at least one iteration of the following steps: - for each defect: - obtaining (40 ) data via said set of sensors positioned according to a current location, - resolution (42) and obtaining a solution of a first low-level optimization problem for reconstructing said fault considered; - determination (44) of a criterion associated with a second high-level optimization problem of the quality of reconstruction of said defects, - determination (46) of the gradient of said criterion, - movement (52) of each sensor of the value of said gradient and obtaining its location for the next iteration. Figure for abstract: Figure 2

Description

Method for bi-level optimization of the location of defect detection sensors in a structure using guided elastic wave tomography

The present invention relates to a method for bi-level optimization of the location of a set of defect detection sensors in a structure by guided elastic wave tomography.

The invention also relates to a computer program comprising software instructions which, when executed by a computer, implement such a method of bi-level optimization of the location of a set of fault detection sensors (s) in a structure by guided elastic wave tomography.

The invention also relates to an electronic device for bi-level optimization of the location of a set of sensors for detecting fault(s) in a structure by guided elastic wave tomography.

The present invention relates to the field of structural health monitoring or SHM ( Structural Health Monitoring ) aimed at detecting and characterizing, in a planned or periodic manner, damage (ie anomalies) of structures/infrastructure, as well as as that of non-destructive testing (NDT) by measurement on request. Such structural anomalies (ie defects) correspond to modifications in the physical and/or geometric properties of the structure considered likely to affect its performance and/or reliability.

Such a health check of structures, or such a non-destructive test (NDT), are implemented by instrumentation of said structures to be checked, respectively using integrated or removable sensors, for example piezoelectric sensors capable of transmitting and receiving ultrasonic guided elastic waves (OG), and this permanently to monitor their state of health over time or at a given moment.

The permanent integration of sensors on or within structures/infrastructure to monitor their condition makes it possible in particular to guarantee the safety of supporting structures corresponding to thin and/or long mechanical components, such as wind turbine blades, fuselages or components. aircraft engines, metal or composite pipes, tension cables, bridge suspension cables, train rails, or any other structure to be monitored in the “Oil & Gas”, nuclear, aeronautical sector, etc.

In particular, the guided elastic ultrasonic waves emitted by such sensors make it possible to detect structural defects leading to discontinuities or variations in geometry, such as cracks, delamination in composite fuselages, corrosion leading in particular to loss of thickness in metals, and/or erosion etc., at an early stage and thus monitor them for decades.

Such detection is generally well mastered in the laboratory where external effects are limited, controlled and calibrated.

The main challenge associated with such a structural diagnosis is, in real conditions, linked on the one hand to the presence of epistemic uncertainties on the structure or instrumentation such as the positions of the sensors, the properties of the sensors, the properties elastic or geometric of the structure, and on the other hand linked to the presence of unsupervised external influence effects (i.e. unknown) evolving with distinct temporal dynamics.

As an alternative to traditional approaches by reference state, that is to say a reference measurement in the absence of a fault then by comparing the current state to the reference state, in particular by subtraction, correlation, etc. while assuming that the only difference between reference and current states can only be attributed to the presence of a fault, but that all other usage parameters, apart from at best a single rapid effect parameter such as the temperature, are equal between these two reference and current states, imaging reconstruction processes, in particular of the guided elastic wave tomography (OG) type, currently make it possible to reconstruct, via an iterative inversion algorithm, from the results provided by said sensors, map the thickness of the structure and therefore image potential damage/defects.

Tomography by guided elastic waves (OG) is based in particular on wave propagation models, for example, radiation tomography, time-of-flight tomography, as described in particular by JC P McKeon and MK Hinders in the article entitled “ Parallel projection and crosshole Lamb contact scanning tomography ” from 1999 and diffraction tomography, as described in particular by F. Simonetti and H. Huang in the article entitled “ From beamforming to diffraction tomography ” from 2008.

In addition, thanks to autocalibration methods described in particular by T. Druet in the article entitled “ Autocalibration method for guided wave tomography with undersampled data ” from 2019, tomography has the potential to be more robust than state-based approaches. reference since the reconstruction of faults is made only from the current state (ie signals measured on a structure already presenting a potential fault).

However, in tomography, theoretical criteria are used to determine the number of sensors necessary in theory to achieve an optimal defect reconstruction, a number which is most often very large and greater than that required by conventional reference state approaches, and therefore prohibitive for many SHM applications, particularly in terms of cost, added mass and/or integration complexity.

Indeed, as a non-limiting example, in a laboratory environment and to control an area of the order of m², classic approaches by reference state require around ten sensors, while tomography methods require of the order of a hundred sensors (to achieve the maximum resolution and precision allowed by the tomography method.

Thus, the main pitfall of tomography by guided elastic waves (OG) is the number of sensors required and the maximum theoretical resolution, considering for example that a spatial resolution of half a wavelength would be achieved with one hundred and twenty sensors (in other words, considering an inter-sensor distance less than or equal to half a wavelength, the minimum number of sensors is equal to the perimeter of the area to be inspected divided by said inter-sensor distance).

In practice, to use tomography by guided elastic waves (OG), the number of sensors used is often voluntarily reduced to respect the constraints of cost, added mass and/or complexity of the SHM system stated previously, but this obviously leads in return to degraded reconstructions of defects.

To remedy this, tomography methods coupled with compressed acquisition (CS) have been developed as described in particular by M. Chang et al. in the article entitled “ Corrosion monitoring using a new compressed sensing-based tomographic method ” from 2020, and are based on the implementation of signal processing (ie data processing) specific to reconstructing data sampled under the frequency of Nyquist (ie respecting the criterion of S, which specifically amounts in SHM to reconstructing an image by tomography using a reduced and limited number of piezoelectric sensors capable of emitting and receiving guided elastic waves (OG), while retaining qualities maximum fault reconstructions.

However, these latest developments generally feature even lower resolution than that associated with diffraction tomography.

Furthermore, in compressed acquisition (CS) it is known that a random/inconsistent measurement process is more favorable to reconstruction than regular sampling.

The aim of this invention is therefore to propose a method and a device capable of providing an early and reliable diagnosis of structural defect in real conditions of use, freeing itself from a reference state (i.e. absolutely), while by being robust and providing the most faithful imaging of defects possible, while limiting the number of sensors necessary to maintain maximum reconstruction quality, particularly in the context of diffraction tomography.

To this end, the subject of the invention is a method for bi-level optimization of the location of a set of defect detection sensors in a structure by guided elastic wave tomography, the sensors of said set being arranged on said structure,

said method comprising a bi-level optimization, said bi-level optimization comprising:

- a phase of initialization of a reference detection configuration, and generation of a database respectively comprising a predetermined number N of distinct reference images of maps of the thickness of said structure respectively presenting N distinct defects from one image to another, and

until convergence according to a predetermined stopping criterion, at least one iteration of the following successive steps:

- for a current location of said set of sensors associated with said iteration, and for each of said N distinct faults, obtaining data suitable for being measured by said set of sensors positioned according to said location,

- from said data, and for each of said N distinct defects, resolution and obtaining a solution of a first low-level optimization problem of reconstruction by tomography of said defect considered, said solution corresponding to the reconstructed image of the mapping of the thickness of said structure presenting said defect considered;

- from said N reconstructed images associated respectively with each of said N distinct defects and from the N reference images of the database associated respectively with each of said N distinct defects, determination of a predetermined criterion associated with a second problem of high-level optimization of the reconstruction quality of said N distinct defects,

- determination of the gradient of said predetermined criterion associated with the second optimization problem,

- displacement of each sensor of the value of said gradient of said criterion associated with each sensor and obtaining the location of each sensor to be used during the following iteration.

This process thus aims to learn, by means of a bi-level optimization, the placement of the set of sensors which makes it possible to best reconstruct the N defects correctly characterized beforehand within the database.

In other words, during the generation of the database the mapping of the thickness of the structure to be inspected is rigorously obtained for each of the N defects, by precise simulation of each defect, and the method according to the present invention aims to learn at as the iterations progress, the best positioning compromise for all of the sensors which makes it possible to obtain, for all of the N distinct defects, all of the N images reconstructed by tomography best approximating the N reference images of the database respectively associated with each distinct fault.

In other words, the method according to the present invention amounts to considering the position (i.e. the location) of all the sensors as a parameter of the bi-level optimization.

According to other advantageous aspects of the invention, the method of bi-level optimization of the location of a set of defect detection sensors in a structure by guided elastic wave tomography comprises one or more of the characteristics following, taken individually or in all technically possible combinations:

- the initialization of the reference detection configuration includes the definition of the elements belonging to the group comprising at least the following elements:

- the size of the area of the structure to be inspected by means of said detection;

- the wavelength of interest;

- the number of sensors theoretically necessary for said detection;

- the number of sensors used in practice;

- the initial positioning of said sensors;

- said initial positioning of said sensors corresponds to a regular circular arrangement of said sensors;

- said initialization phase further comprises obtaining an asymmetry of said initial positioning by applying a displacement of predetermined value in a random direction of each of said sensors;

- the determination of the gradient of said criterion includes:

for each of said N distinct defects:

- determining a double gradient of said first low-level optimization problem, by differentiating the gradient of said first low-level optimization problem as a function of its parameters;

- determining an associated Hessian matrix as a function of the data respectively from said database, said Hessian matrix being invertible,

- from said Hessian matrix, the determination of a Jacobian matrix according to a predetermined obtaining relation; And

- from the set of Jacobian matrices respectively associated with each of said N distinct defects, determination of the gradient of said criterion.

- said predetermined stopping criterion corresponds to:

- a Euclidean norm of the gradient of the high-level problem lower than a predetermined threshold; Or

- reaching a predetermined maximum number of iterations.

- said predetermined threshold corresponds to a predetermined fraction of the wavelength of interest initialized during the initialization phase of said reference detection configuration;

- said database can be reconfigured according to information on a type of fault a priori to be detected.

The invention also relates to a computer program comprising software instructions which, when executed by a computer, implement such an optimization method by learning the location of a set of defect detection sensors(s) in a structure by guided wave tomography as defined above.

The invention also relates to an optimization device by learning the location of a set of sensors for detecting fault(s) in a structure by guided wave tomography, the sensors of said set being arranged on said structure, said device comprising a bi-level optimization unit, said bi-level optimization unit level including:

- an initialization module configured to initialize a reference detection configuration, and configured to generate a database comprising respectively a predetermined number N of distinct reference images of maps of the thickness of said structure respectively presenting N distinct defects from one image to another, and

implemented iteratively; until convergence according to a predetermined stopping criterion, the following elements:

- an obtaining module configured, for a current location of said set of sensors associated with said iteration, and for each of said N distinct defects, obtaining data suitable for being measured by said set of sensors positioned according to said location,

- a resolution module configured to resolve and obtain, from said data, and for each of said N distinct defects, a solution of a first low-level optimization problem of reconstruction by tomography of said defect considered, said solution corresponding to the reconstructed image of the thickness mapping of said structure presenting said defect considered;

- a first determination module configured to determine, from said N reconstructed images associated respectively with each of said N distinct defects and from the N reference images from the database associated respectively with each of said N distinct defects, an associated predetermined criterion to a second high-level optimization problem of the reconstruction quality of said N distinct defects,

- a second determination module configured to determine the gradient of said criterion,

- a movement module configured to move each sensor by the value of said gradient of said criterion associated with each sensor and to obtain the location of each sensor to be used during the following iteration.

The invention also relates to a system for detecting anomaly(s) in a structure by guided wave tomography by means of a set of sensors, the sensors of said set being arranged on said structure, said system being comprising:

- said set of sensors;

- an electronic device for optimizing the placement of said set of sensors as described previously;

- a guided wave tomography anomaly detection module configured to use the measurements of said set of sensors in order to provide a tomography of said structure.

According to other advantageous aspects of the invention, the system for detecting anomaly(s) in a structure by guided wave tomography by means of a set of sensors comprises one or more of the following characteristics, taken in isolation or according to all technically possible combinations:

- the sensors of said assembly are according to one of the types belonging to the group comprising at least:

- piezoelectric sensors;

- EMAT sensors;

- FBG sensors;

- PVDF sensors;

- the system is capable of operating in:

- active mode in which the sensors generate and measure the guided waves, or in

- passive mode according to which the sensors are configured to measure the guided waves present in the structure naturally during its operation, the signals measured by said sensors in passive mode being suitable for being used to determine a function representative of the impulse response of said structure;

- the guided wave tomography anomaly detection module configured to use the measurements of said set of sensors in order to provide a tomography of said structure comprises a solver allowing the resolution of a minimization problem in the form:

Or represents the object function characterizing a defect in said structure, R is a penalty term allowing the regularization of the solution, represents the fields measured via said sensors of said assembly, represents a model following a Lippmann Schwinger equation:

being Green's function, solution of the Helmholtz equation associated with the healthy state of said structure

These characteristics and advantages of the invention will appear more clearly on reading the description which follows, given solely by way of non-limiting example, and made with reference to the appended drawings, in which:

- there is a diagram illustrating a device for optimizing the placement of a set of defect detection sensors in a structure by guided wave tomography according to the present invention;

- there is a flowchart of a bi-level optimization method for the location of a set of fault detection sensors in a structure by guided elastic wave tomography according to the present invention;

- there is a diagram generally illustrating the operation of guided wave tomography;

- there illustrates the movement of sensors obtained after bi-level optimization from an initial configuration;

- there compares on a given example the results obtained according to the present invention with those obtained according to the state of the art.

In the remainder of the description, the expression "substantially equal to" is understood as a relation of equality to plus or minus 10%, that is to say with a variation of at most 10%, preferably again as a relationship of equality at plus or minus 5%, that is to say with a variation of at most 5%.

There is a representation of an electronic device 10 for bi-level optimization of the location of a set of fault detection sensors in a structure by guided elastic wave tomography, the sensors (not shown) of said set being arranged on said structure (not shown on the ).

It should be noted that said sensors are able to operate in “active mode” according to which the sensors generate and measure the guided waves (e.g. ultrasonic), or in “passive mode” according to which the sensors simply measure the guided waves (e.g. ultrasound) present in the structure naturally during its operation.

Passive methods (noise correlation, passive inverse filter or even correlation coda correlation) make it possible to reconstruct the same signals (or practically the same depending on whether certain assumptions are respected) as those measured in active mode but in a completely passive manner.

More precisely, there is no emission of waves by the sensors in passive mode. The sensors only measure the ambient noise, over a sufficiently long time, naturally present in the structure inspected during its operation. Ambient noise is a sum of guided waves of different intensities and propagating in different directions. This ambient noise is then processed by a so-called passive method (i.e. passive processing), examples of which are described in patent applications FR 3 073 289, FR 3 084 748, FR 3 105 554, making it possible to determine a function representative of the impulse response of the structure (corresponding substantially to the signal measured in active mode). Such signal reconstructions are suitable for use as tomography input data in the same manner as in active mode.

In addition, by sensors, we hereinafter mean sensors according to one of the following types of sensors or combinations of sensors: piezoelectric sensors, Bragg gratings on optical fiber, or FBG (from the English Fiber Bragg Gratings ) , EMAT (from the English Electro magneto acoustic transducer ), PVDF (Polyvinylidene fluoride), etc.

According to the present invention, the device 10 comprises a bi-level optimization unit (not shown as such) comprising first of all an initialization module 12 configured to initialize a reference detection configuration, and configured to generate a database comprising respectively a predetermined number N of distinct reference images mapping the thickness of said structure respectively presenting N distinct defects from one image to another.

According to a complementary optional variant, described in more detail below in relation to the illustrating the method implemented by said electronic device 10, the initialization module 12 is more precisely configured to define elements belonging to the group comprising at least the following elements:

- the size of the area of the structure to be inspected by means of said detection;

- the wavelength of interest ;

- the number of sensors theoretically necessary for said detection;

- the number of sensors used in practice;

- the initial positioning of said sensors.

Furthermore, according to the present invention, the bi-level optimization unit of the electronic device 10 also comprises a set of modules capable of being implemented successively during the same iteration; each iteration being repeated until convergence according to a predetermined stopping criterion.

Such a set of modules firstly comprises an obtaining module 14 configured to obtain, in particular by simulation, for a current location of said set of sensors associated with said iteration, and for each of said N distinct defects, data capable of being measured by said set of sensors positioned according to said location.

Furthermore, according to the present invention, this set of modules further comprises a resolution module 16 configured to resolve and obtain, from said data provided by the obtaining module 14, and for each of said N distinct defects, a solution of a first low-level optimization problem of reconstruction by tomography of said defect considered, said solution corresponding to the reconstructed image of the thickness mapping of said structure presenting said defect considered.

Furthermore, according to the present invention, the bi-level optimization unit of the electronic device 10 also comprises a first determination module 18 configured to determine, from said N reconstructed images associated respectively with each of said N distinct defects and from N reference images from the database associated respectively with each of said N distinct defects, a predetermined criterion associated with a second high-level optimization problem of the reconstruction quality of said N distinct defects.

Furthermore, according to the present invention, the bi-level optimization unit of the electronic device 10 also comprises a second determination module 19 configured to determine the gradient of said criterion.

Finally, according to the present invention, the bi-level optimization unit of the electronic device 10 also comprises a displacement module 20 configured to move each sensor by the value of said gradient of said criterion associated with each sensor and to obtain the location of each sensor to use in the next iteration.

In the example of the , the electronic device 10 for bi-level optimization of the location of a set of defect detection sensors in a structure by guided elastic wave tomography comprises an information processing unit 22 formed for example of a memory 24 and a processor 26 associated with the memory 24.

In the example of the , the initialization module 12, the obtaining module 14, the resolution module 16, the first determination module 18, the second determination module 19, and the movement module 20 are each produced in the form of software , or a software brick, executable by the processor 26. The memory 24 of the electronic device 10 for bi-level optimization of the location of a set of fault detection sensors (s) in a structure by wave tomography guided elastics is then capable of storing, to implement said bi-level optimization, initialization software, and iteratively, obtaining software, resolution software, first determination software, second determination software. determination, and movement software. The processor 26 is then able to execute each of the software programs among the initialization software, the obtaining software, the resolution software, the first determination software, the second determination software, and the movement software.

In a variant not shown, the initialization module 12, the obtaining module 14, the resolution module 16, the first determination module 18, the second determination module 19, and the movement module 20 are each produced in the form of a programmable logic component, such as an FPGA ( Field Programmable Gate Array ), or a GPU graphics processor ( Graphics Processing Unit ), or even in the form of an integrated circuit, such as an ASIC ( Application Specific Integrated Circuit ).

When the electronic device 10 for bi-level optimization of the location of a set of defect detection sensors in a structure by guided elastic wave tomography is produced in the form of one or more software programs, it is that is to say in the form of a computer program, also called a computer program product, it is also capable of being recorded on a medium, not shown, readable by computer. The computer-readable medium is, for example, a medium capable of storing electronic instructions and of being coupled to a bus of a computer system. For example, the readable medium is an optical disk, a magneto-optical disk, a ROM memory, a RAM memory, any type of non-volatile memory (for example EPROM, EEPROM, FLASH, NVRAM), a magnetic card or an optical card. A computer program comprising software instructions is then stored on the readable medium.

The operation of the electronic device 10 for bi-level optimization of the location of a set of fault detection sensors in a structure by guided elastic wave tomography will now be described with reference to the which schematically illustrates an example of implementation, according to the present invention, of a method 28 for bi-level optimization of the location of a set of defect detection sensors in a structure by elastic wave tomography guided, the sensors of said assembly being arranged on said structure (not shown).

More precisely, according to the present invention, the method 28 corresponds to a bi-level optimization of the location of the set of detection sensors comprising first of all a step 30 of initializing a reference detection configuration, and generating a database comprising respectively a predetermined number N of distinct reference images mapping the thickness of said structure respectively presenting N distinct defects from one image to another.

As an optional complement, said initialization 30 of the reference detection configuration includes the definition of the elements belonging to the group comprising at least the following elements:

- the size of the area of the structure to be inspected by means of said detection;

- the wavelength of interest;

- the number of sensors theoretically necessary for said detection;

- the number of sensors used in practice;

- the initial positioning of said sensors.

To do this, according to the example of Figure 2, the initialization phase 30 comprises a first sub-step 31 of choice C_C of the reference configuration as such in which the size of the zone to be inspected, the length d wave of interest , the number of sensors necessary in theory to inspect said area, and the number of sensors chosen by the users are defined, in particular by means of a user interface not shown on the .

According to this optional complement, the initialization phase 30 further comprises a sub-step 32 for defining and generating the database, called the training database, B_E. When generating the B_E database, the mapping of the thickness of the structure to be inspected is rigorously obtained for each of the N defects, by precise simulation of each defect. According to one example, N is notably equal to fifty distinct defects (in size, shape, etc.) simulated precisely to obtain for each of the N defects, the mapping of the thickness of the structure to be inspected with the defect considered.

According to this optional complement, the initialization phase 30 further comprises a sub-step 33 of initial positioning of said sensors, said initial positioning P _i being the starting point for the iterative implementation of the iterative bi-level optimization in as such and as described below.

As an optional complement, said initial positioning P _i of said sensors corresponds to a regular circular arrangement of said sensors.

As an optional complement, said sub-step 33 of the initialization phase 30 further comprises obtaining an asymmetry of said initial positioning by applying a displacement of predetermined value in a random direction of each of said sensors.

As an optional complement, said sub-step 32 of defining and generating the so-called training database, B_E, is repeatable so as to make said database B_E reconfigurable according to information on an a priori type of fault. to detect.

Then, the method 28 for bi-level optimization of the location of a set of defect detection sensors in a structure by guided elastic wave tomography comprises an iterative loop repeated until convergence according to a criterion of predetermined stop C _A (ie convergence criterion) tested during step 34.

As long as the predetermined stopping criterion (ie convergence criterion) C _A is not reached as illustrated by arrow 36, the iterative loop 38 is implemented.

The iterative loop 38 comprises the five successive steps described below, namely first of all a step 40 of obtaining data suitable for being measured by said set of sensors positioned according to a current location of said set of sensors associated with said current iteration, and for each of said N distinct faults used beforehand during the initialization phase 30 to generate the training base B_E.

Then, the iterative loop 38 comprises, from said data, and for each of said N distinct defects, a step 42 of resolving R_P ₁ and obtaining a solution of a first low-level optimization problem of reconstruction by tomography of said defect considered, said solution corresponding to the reconstructed image of the thickness mapping of said structure presenting said defect considered.

Then, the iterative loop 38 comprises, from said N reconstructed images associated respectively with each of said N distinct defects and from the N reference images from the database associated respectively with each of said N distinct defects, a step 44 of determining a predetermined C_HN criterion associated with a second high-level optimization problem of the quality of reconstruction of said N distinct defects.

Then, the iterative loop 38 comprises a step 46 of determining the gradient G_C_HN of said predetermined criterion C_HN associated with the second optimization problem.

More precisely, as an optional complement as illustrated by the , the determination 46 of the gradient G_C_HN of said criterion C_HN includes:

for each of said N distinct defects:

- a substep 47 for determining a double gradient DG of said first low-level optimization problem, by differentiating the gradient of said first low-level optimization problem as a function of its parameters;

- a substep 48 for determining an associated Hessian matrix H as a function of the data respectively from said database, said Hessian matrix H being invertible,

- from said Hessian matrix, a sub-step 49 of determining a Jacobian matrix J according to a predetermined obtaining relation; And

- from the set of Jacobian matrices respectively associated with each of said N distinct defects, a substep 50 for determining CG_C_HN of the gradient of said criterion.

Finally, the iterative loop 38 includes a step 52 of moving each sensor by the value of said gradient of said criterion associated with each sensor and of obtaining the location of each sensor to be used during the following iteration illustrated by arrow 54.

As represented by the arrow 54, it is therefore an iterative method where we repeat (ie reiterate) successively the steps 40, 42, 44, 46, 52 until convergence in order to optimize the placement of the sensors. Once the convergence criterion has been reached as tested in step 34, we obtain, according to arrow 56, as a result 58 a positioning P _opt deemed "optimal" of the sensors corresponding to the local minimum of the high level criterion corresponding to the aforementioned initialization 30 of the position of the sensors in particular, (two distinct initialization configurations being able to lead respectively to two distinct local minimums (ie the method according to the present invention is not deterministic because depending on the initialization the local minimum found by the process may be different).

Subsequently, the steps of the method 28 are described in more detail in relation to Figures 3 to 5.

In particular, the is a diagram generally illustrating how guided wave tomography works.

More precisely, on the “real” schematic view 60 of the , a supporting structure, namely a plate, of a defect 62 is shown instrumented by a network of sensors 64, for example piezoelectric. Such a defect 62 likely to be present in the structure corresponds for example to a loss of thickness by corrosion/erosion, or to delamination in a composite or even to a crack in a metallic material.

On the , the symbol 66 represents the generation/measurement, by the sensors 64, of ultrasonic waves. View 68 then illustrates, after extraction and calibration of the fields φ of the acquired time signals, the processing of the signals obtained, and in particular the representation of the theoretical flight time associated with the signal acquired by each sensor 64 (ie each sensor 64 implements data acquisition).

From these fields acquired by the sensors 64, the reconstruction of a speed map of the area to be inspected (ie the tomography itself) is implemented and based on a model following the Lippmann Schwinger equation:

(1)

being Green's function, solution of the Helmholtz equation associated with the healthy state of the structure.

Such a reconstruction is for example carried out using a solver, for example non-linear (or as an alternative using a succession of linear iterations), allowing the resolution of a minimization problem in the form:

. (2)
Or represents, as indicated previously, the measured fields (ie data ) via said sensors 64, represents the aforementioned model according to the Lippmann Schwinger equation, the object function characterizing the defect 62, and R is a penalty term allowing the regularization of the solution. It is therefore a question of finding the object function which achieves the best compromise between a measurement fidelity term depending on model and a regular structure imposed by R.

Note that if the regularization R is proportional to the L1 norm, such a reconstruction is in a “Lasso” type configuration as described by Q. Bertrand et al. in the article entitled “Implicit differentiation of Lasso-type models for hyperparameter optimization” from 2020.

Such a reconstruction by tomography is suitable for being coupled with a compressed acquisition (CS for Compressed Sensing ), making it possible to reconstruct signals sampled under the Nyquist frequency.

According to the application of the present invention, the subsampling is spatial (i.e. sampled in space, and not temporal), and to function, the compressed acquisition CS requires compliance with two key principles, namely on the one hand the parsimony of the data (i.e. the unknown has a small number of non-zero coefficients) or the compressibility of the data (i.e. the unknown can be described with few coefficients in a well-chosen mathematical basis), and on the other hand the inconsistency of the measurement process, for example obtained through random sampling, the present invention seeking to maximize the inconsistency (i.e. minimize the coherence) of the measurement process by placing the sensors in a clever manner , by bi-level optimization as described in detail below, the reconstruction problem becoming the first low-level problem.

Once the reconstruction of the speed map has been carried out, this speed map is converted into a thickness map, as illustrated by view 70 thanks to the dispersive properties of the guided waves. In particular, on view 70, the wave number space (i.e. the Ewald circle) of placement of the sensors 72 is represented, as is the zone 74 of the thickness map corresponding to the defect 62 of the real view 60.

It should be noted that the present invention aims above all to optimize the placement of the sensors 64, and applies to different variants of reconstruction by guided wave tomography, in particular by means of different regularization functions.R(.), preferably an iterative diffraction tomography reconstruction algorithm as described by F. Simonetti et al. in the article entitled« From beamforming to diffraction tomography» of 2008, or by time-of-flight tomography as previously cited.

View 70 corresponds in particular to the reconstruction by classic tomography using a regular circular network of thirty sensors to notably respect the criterion of Shannon Nyquist which theoretically makes it possible to obtain a perfect classic tomographic reconstruction but at the cost of generally requiring a large number of sensors. Indeed, in classic guided wave tomography, to achieve maximum defect reconstruction quality, it is necessary to place a sensor every half wavelength.

Below, the bi-level optimization according to the present invention is described in more detail.

It should be noted that Q. Bertrand et al. in the article entitled “Implicit differentiation of Lasso-type models for hyperparameter optimization” from 2020 generally describes a bi-level optimization applied to the general case of mathematical estimation of the best regularization parameter associated with a Lasso-type problem, and not applied, as proposed according to the present invention and described subsequently, to guided wave tomography in the field of structural health monitoring or SHM ( Structural Health Monitoring ).

The general and mathematical context of this document poses a Lasso type problem in which a vector x of minimum L1 norm, connected to a measurement vector y by a model A is sought, and with a set of parameters given, defines the following criterion:

(3)

the vector x of minimum L1 norm retained then corresponding to the approximation expressed by means of the accent ^: . This first reconstruction problem (associated with the lower index) is subsequently designated as the low-level problem. Then Q. Bertrand et al. proposes to evaluate the reconstruction performance of the algorithm on a training base of N known images and an associated dataset for the index i varying from 1 to N.

Still according to this article, the best regularization parameter is a solution to a problem of the following type:

(4)

where L defines a distance between known images and estimated images and P is a penalty making it possible to optionally regularize λ . This problem of determining the best regularization parameter is subsequently referred to as a high-level problem.

Solving two optimization problems nested within each other defines what is called bi-level optimization.

According to this article by Q. Bertrand et al., the technical difficulty of such a bi-level optimization is associated with the implementation of a gradient descent of the high-level problem. Indeed, calculating the gradient of L as a function of λ : requires knowing the expression of the Jacobian matrix generically comprising the partial derivatives of each of the components (k) of the image estimated as a function of each parameter (l), and proposes to overcome it the postulate that the solution of the low-level problem cancels the gradient of the low-level problem: (5), which then makes it possible to differentiate this relationship according to the parameters, to obtain:

(6), to finally obtain the following expression of the Jacobian, provided that the Hessian matrix is invertible: (7).

In other words, the article by Q. Bertrand et al. proposes a general and mathematical method of differentiation applicable for the optimization of parameters of a reconstruction algorithm in the context of a bi-level optimization.

The present invention proposes to adapt, judiciously and specifically, the general principle described in the article by Q. Bertrand et al. to guided wave tomography in the field of structural health monitoring or SHM (from English Structural Health Monitoring ), by cleverly (and non-obviously) considering the position sensors as parameters to be optimized (instead of the parameter from the article by Q. Bertrand et al).

The bi-level optimization approach proposed according to the present invention is therefore applied specifically to guided wave tomography, and is also composed of two optimization problems nested one inside the other, these two problems being distinct.

More precisely, the first level, called low level, is, specifically according to the present invention, the tomographic reconstruction as described in the following equation, corresponding to a rewriting of equation (2) previously cited:

(8)

where s is the so-called sparse representation of the object function , represents the deviation from the data to be reconstructed, represents the set of parameters, designates the underdetermined problem to be reversed with the measurements φ obtained during step 40 previously mentioned of method 38 according to the present invention, the model following the aforementioned Lippmann Schwinger equation (1), and R is the regularization penalty.

In other words, during step 40, placement of given sensors, the data are generated digitally for each fault i of the base B_E using the known model .

Then, during step 42, each low-level problem to obtain the solutions , following equation (8) is solved.

The second level problem, or high -level problem, can be formulated as follows:

(9)

and corresponds to the optimization of the quality of the reconstruction on the set of N test faults previously used to generate the training database B_E, corresponding to what is called according to the present invention the high-level criterion.

In other words, during step 44, the reconstruction performance is evaluated by evaluating said high-level criterion .

During step 46, sub-step 47 corresponds to the determination of the double gradient of said first low-level optimization problem (low) , by differentiating the gradient of said first low-level optimization problem as a function of its parameters, in other words, the differentiation of the measurements and the model depending on the parameters to be optimized, the low-level problem being assumed to be linear, which implies that the Born approximation is respected.

During substep 48, and always for each of said N distinct defects of index i , the Hessian matrix is determined by calculation from the associated data (ie s being a so-called parsimonious representation of the object function ), said Hessian matrix being invertible.

During substep 49, and still for each of said N distinct defects of index i, the Jacobian is determined by calculation associated by applying equation (7) previously mentioned.

During sub-step 50, the gradient of the high-level problem is determined by calculation from the different Jacobians calculated previously during substep 49.

In other words, the method 38 according to the present invention proposes to optimize the parameters of the reconstruction on a training database, which amounts to learning whose final quality depends on the richness and representativeness of the learning basis.

Cleverly, the method 38 according to the present invention amounts to considering all the positions of the sensors in the set of parameters as a parameter of the optimization system in the same way as the regularization parameter λ in the general Lasso approach in the article by Q. Bertrand et al.

The movement of the sensors is capable of acting both on the operator representing the direct model (instead of model A in the article by Q. Bertrand et al.) and on the measurements (instead of the y measurements in the article by Q. Bertrand et al.) which is taken into account when calculating the Jacobian matrix J.

The movement of the sensors resulting from the optimization of the high level problem thus makes it possible to reduce the practical coherence of the measurement process, which benefits the solver to solve the low level problem.

As previously indicated in relation to the , the bi-level optimization method 38 according to the present invention is iterative and as an optional complement the predetermined stopping criterion C _A corresponds to:

- a Euclidean norm of the gradient of the high-level problem less than a threshold predetermined, as for example expressed according to the following equation ; Or

- reaching a predetermined maximum number of iterations.

According to an optional variant, said predetermined threshold corresponds to a predetermined fraction of the wavelength of interest initialized during the initialization phase of said reference detection configuration. For example, is determined to define a minimum performance difference for the elementary displacement of the sensor (that is to say for example of the order of a tenth of the wavelength of interest ).

Reaching a predetermined maximum number of iterations is a stopping criterion implemented in particular for safety reasons to limit the duration and computing resources necessary for gradient descent according to method 38 of the present invention.

In practice, the inventors have observed that the movement of the sensors tends to bring them closer to the defects, but in the context of guided wave tomography, a “wide” zone to be inspected in which the defects are likely to appear is generally defined. , and the present invention makes it possible to constrain the movement of the sensors to a ring, the average radius of which is the radius of the circular reference distribution associated with the initial positioning P _i fixed during the initialization phase 30 and its thickness is l 'order of the wavelength of interest .

There illustrates the movement in the space of wavenumbers 80, of a configuration obtained after bi-level optimization from an initial configuration with fifteen sensors.

In Figure 4, the initial position 82 of the sensors is represented as well as the new associated position 84, the lines 86 representing the translation (ie the movement) of each sensor, the optimal positioning 84 having been learned using a database including forty images of simple bowl-shaped corrosion type defects. We notice in Figure 4 that the movements of the sensors tend to get closer to the center of the area to be inspected. In addition, the movement of the sensors is of the order of magnitude of the wavelength .

There compares on a given example the results obtained according to the present invention with those obtained according to the state of the art (ie positioning of regular sensors) by quantifying the reconstruction errors according to the selected method.

On view 90, the reconstruction error observed in simulation for regularly positioned sensors is represented by means of points 92 for configurations with ten and twelve sensors and serves as a point of comparison. The reconstruction error of a simple fault for five configurations with ten sensors and five configurations with twelve sensors observed in simulation for the sensors positioned by means of the bi-level optimization according to the present invention is represented by means of points 94 of same as their average performance by means of 96 points.

First of all, we note that, resulting from a different initialization, the 2 X 5 final configurations at the output of the bi-level optimization method according to the present invention are local minima different from a criterion evaluated on the same basis , the observed criterion being in the general case non-convex. Furthermore, such minimums on the training base do not necessarily guarantee better performance (lower error) than in the regular case associated with point 92. However, we note that it is possible to obtain better performance than the regular network and that the average behavior seems to give better results when working with only ten sensors, which is consistent with an approach to minimizing the number of sensors. Thus the method proposed according to the present invention is capable of making it possible to learn the ideal positioning of the sensors in specific cases as long as the training base is representative of the faults to be reconstructed.

Those skilled in the art will understand that the invention is not limited to the embodiments described, nor to the particular examples of the description, the embodiments and the variants mentioned above being suitable for being combined with each other to generate new embodiments of the invention.

Thus, the present invention proposes a method and a device for bi-level optimization of the location of a set of fault detection sensors in a structure by guided elastic wave tomography which makes it possible to cleverly determine the placement of the sensors by learning, by means of a bi-level optimization, the placement of the set of sensors which makes it possible to best reconstruct the N faults correctly characterized beforehand within the database

This learning technique has an advantage in the fact that it can be adaptive to the types of defects that we seek to reconstruct. For example, the optimal positioning of sensors to analyze cracks appearing horizontally would not be the same as that for analyzing approximately circular corrosion defects. Any a priori information can therefore make it possible to improve performance via the learning and testing database (i.e. the training database).

The invention makes it possible to cleverly choose the placement of sensors to maximize the information from each one while limiting their number. Indeed, in conventional guided wave tomography, to achieve maximum defect reconstruction quality, it is necessary to place a sensor every half wavelength. By doing this, the proposed solution allows a given number of sensors to be placed to optimize fault reconstruction performance on a fault basis. Any a priori information transmitted by the user for the creation of the training base (for example: size, shape, positioning of defects, etc.) makes it possible to increase the performance of the present invention, which is advantageous for applications in the field of structural health control or SHM (from the EnglishStructural Health Monitoring) aimed at detecting and characterizing damage (i.e. anomalies) of structures/infrastructure such as metal or composite pipes for nuclear power, the “Oil & Gas” sector, aeronautics, wind power, etc. .

Claims (14)

Method (28) for bi-level optimization of the location of a set of defect detection sensors in a structure by guided elastic wave tomography, the sensors of said set being arranged on said structure,
said method being characterized in that it comprises a bi-level optimization, said bi-level optimization comprising:
- a phase of initialization (30) of a reference detection configuration, and generation of a database respectively comprising a predetermined number N of distinct reference images of maps of the thickness of said structure respectively presenting N distinct defects from one image to another, and
until convergence according to a predetermined stopping criterion, at least one iteration (38) of the following successive steps:
- for a current location of said set of sensors associated with said iteration, and for each of said N distinct faults, obtaining (40) data suitable for being measured by said set of sensors positioned according to said location,
- from said data, and for each of said N distinct defects, resolution (42) and obtaining a solution of a first low-level optimization problem of reconstruction by tomography of said defect considered, said solution corresponding to the image reconstructed map of the thickness of said structure presenting said defect considered;
- from said N reconstructed images associated respectively with each of said N distinct defects and from the N reference images of the database associated respectively with each of said N distinct defects, determination (44) of a predetermined criterion associated with a second high-level optimization problem of the reconstruction quality of said N distinct defects,
- determination (46) of the gradient of said predetermined criterion associated with the second optimization problem,
- displacement (52) of each sensor of the value of said gradient of said criterion associated with each sensor and obtaining the location of each sensor to be used during the following iteration. Method (28) according to claim 1, wherein the initialization (30) of the reference detection configuration comprises the definition of the elements belonging to the group comprising at least the following elements:
- the size of the area of the structure to be inspected by means of said detection;
- the wavelength of interest;
- the number of sensors theoretically necessary for said detection;
- the number of sensors used in practice;
- the initial positioning of said sensors. A method (28) according to claim 2, wherein said initial positioning of said sensors corresponds to a regular circular arrangement of said sensors. Method (28) according to claim 3, wherein said initialization phase further comprises obtaining an asymmetry of said initial positioning by applying a displacement of predetermined value in a random direction of each of said sensors. Method (28) according to any one of the preceding claims, in which the determination (46) of the gradient of said criterion comprises:
for each of said N distinct defects:
- determining (47) a double gradient of said first low-level optimization problem, by differentiating the gradient of said first low-level optimization problem as a function of its parameters;
- determining (48) an associated Hessian matrix as a function of the data respectively from said database, said Hessian matrix being invertible,
- from said Hessian matrix, determining (49) a Jacobian matrix according to a predetermined obtaining relation; And
- from the set of Jacobian matrices respectively associated with each of said N distinct defects, determination (50) of the gradient of said criterion. Method (28) according to any one of the preceding claims, in which said predetermined stopping criterion corresponds to:
- a Euclidean norm of the gradient of the high-level problem lower than a predetermined threshold; Or
- reaching a predetermined maximum number of iterations. Method (28) according to claim 6, wherein said predetermined threshold corresponds to a predetermined fraction of a wavelength of interest initialized during the initialization phase of said reference detection configuration. Method (28) according to any one of the preceding claims, in which said database is reconfigurable as a function of information on an a priori type of fault to be detected. Computer program comprising software instructions which, when executed by a computer, implement a bi-level optimization method for the location of a set of fault detection sensors in a structure by tomography with guided elastic waves, according to any one of the preceding claims. Device (10) for bi-level optimization of the location of a set of defect detection sensors in a structure by guided elastic wave tomography, the sensors of said set being arranged on said structure, said device being characterized in that it comprises a bi-level optimization unit, said bi-level optimization unit comprising:
- an initialization module (12) configured to initialize a reference detection configuration, and configured to generate a database comprising respectively a predetermined number N of distinct reference images of maps of the thickness of said structure respectively presenting N distinct defects from one image to another, and
implemented iteratively; until convergence according to a predetermined stopping criterion, the following elements:
- an obtaining module (14) configured, for a current location of said set of sensors associated with said iteration, and for each of said N distinct defects, to obtain data suitable for being measured by said set of sensors positioned according to said location,
- a resolution module (16) configured to resolve and obtain, from said data, and for each of said N distinct defects, a solution of a first low-level optimization problem of reconstruction by tomography of said defect considered, said solution corresponding to the reconstructed image of the thickness mapping of said structure presenting said defect considered;
- a first determination module (18) configured to determine, from said N reconstructed images associated respectively with each of said N distinct defects and from the N reference images from the database associated respectively with each of said N distinct defects, a predetermined criterion associated with a second high-level optimization problem of the reconstruction quality of said N distinct defects,
- a second determination module (19) configured to determine the gradient of said criterion,
- a movement module (20) configured to move each sensor by the value of said gradient of said criterion associated with each sensor and to obtain the location of each sensor to be used during the following iteration. System for detecting anomaly(s) in a structure by guided wave tomography using a set of sensors, the sensors of said set being arranged on said structure, said system being characterized in that it comprises:
- said set of sensors;
- an electronic device for optimizing the placement of said set of sensors according to claim 10;
- a guided wave tomography anomaly detection module configured to use the measurements of said set of sensors in order to provide a tomography of said structure. System according to claim 11, in which the sensors of said assembly are according to one of the types belonging to the group comprising at least:
- piezoelectric sensors;
- EMAT sensors;
- FBG sensors;
- PVDF sensors. System according to claim 11 or 12, in which the system is capable of operating in:
- active mode in which the sensors generate and measure the guided waves, or in
- passive mode according to which the sensors are configured to measure the guided waves present in the structure naturally during its operation, the signals measured by said sensors in passive mode being suitable for being used to determine a function representative of the impulse response of said structure. System according to any one of claims 11 to 13 in which the guided wave tomography anomaly detection module configured to use the measurements of said set of sensors in order to provide a tomography of said structure comprises a solver allowing the resolution of a minimization problem in the form:
Or represents the object function characterizing a defect in said structure, R is a penalty term allowing the regularization of the solution, represents the fields measured via said sensors of said assembly, represents a model following a Lippmann Schwinger equation:

being Green's function, solution of the Helmholtz equation associated with the healthy state of said structure.